Hierarchical Bayesian modeling of multi-region brain cell count data

Reviewer #1 (Public review):

Summary:

This work proposes a new approach to analyse cell-count data from multiple brain regions. Collecting such data can be expensive and time-intensive, so, more often than not, the dimensionality of the data is larger than the number of samples. The authors argue that Bayesian methods are much better suited to correctly analyse such data compared to classical (frequentist) statistical methods. They define a hierarchical structure, partial pooling, in which each observation contributes to the population estimate to more accurately explain the variance in the data. They present two case studies in which their method proves more sensitive in identifying regions where there are significant differences between conditions, which otherwise would be hidden.

Strengths:

The model is presented clearly, and the advantages of the hierarchical structure are strongly justified. Two alternative ways are presented to account for the presence of zero counts. The first involves the use of a horseshoe prior, which is the more flexible option, while the second involves a modified Poisson likelihood, which is better suited to datasets with a large number of zero counts, perhaps due to experimental artifacts. The results show a clear advantage of the Bayesian method for both case studies.

The code is freely available, and it does not require a high-performance cluster to execute for smaller datasets. As Bayesian statistical methods become more accessible in various scientific fields, the whole scientific community will benefit from the transition away from p-values. Hierarchical Bayesian models are an especially useful tool that can be applied to many different experimental designs. However, while conceptually intuitive, their implementation can be difficult. The authors provide a good framework with room for improvement.

Weaknesses:

Alternative possibilities are discussed regarding the prior and likelihood of the model. Given that the second case study inspired the introduction of the zero-inflation likelihood, it is not clear how applicable the general methodology is to various datasets. If every unique dataset requires a tailored prior or likelihood to produce the best results, the methodology will not easily replace more traditional statistical analyses that can be applied in a straightforward manner. Furthermore, the differences between the results produced by the two Bayesian models in case study 2 are not discussed. In specific regions, the models provide conflicting results (e.g., regions MH, VPMpc, RCH, SCH, etc.), which are not addressed by the authors. A third case study would have provided further evidence for the generalizability of the methodology.

Reviewer #2 (Public review):

Summary:

This is a well-written methodology paper applying a Bayesian framework to the statistics of cell counts in brain slices. A sharpening of the bounds on measured quantities is demonstrated over existing frequentist methods and therefore the work is a contribution to the field.

Strengths:

As well as a mathematical description of the approach, the code used is provided in a linked repository.

Weaknesses:

A clearer link between the experimental data and model-structure terminology would be a benefit to the non-expert reader.

Author response:

We thank both reviewers for their considerate reviews. In this provisional response we would like to make a few key points.

Given that we introduced a bespoke likelihood model for the second dataset, Reviewer 1 asks whether "every unique dataset requires a tailored prior or likelihood to produce the best results". Our intention is to advocate for the horseshoe prior model as a 'standard' first analysis for any cell count dataset. If extra knowledge about the data is available, or if any data artefacts are detected, more elaborate likelihoods could be introduced as needed in a follow-up analysis. Our introduction of the zero-inflated Poisson likelihood for the second dataset was one such example, but many alternatives could exist. This iterative approach to model building, sometimes referred to as a `Bayesian workflow' is seen as good practise in Bayesian data analysis literature. In the revised version of the paper, we will try to explain the recommendations and modelling philosophy behind this method while emphasising that tailoring or bespoke modelling is not required for our `standard analysis', what we would regard as the Bayesian replacement for a t-test on counts.

Reviewer 1 notes that "the differences between the results produced by the two Bayesian models in case study 2 are not discussed". We agree that this discrepancy, arising from the specific assumptions of each model is an interesting issue which we should better explore in the paper. In Figure 6 we plotted the actual data values alongside posterior and confidence intervals to explain how the results from the ZIP likelihood and Horseshoe prior compare with those from a t-test. However, our example regions did not highlight cases where differences could be noted between the the two Bayesian models. In the revised version of the paper, we will extend Figure 6 to include further brain regions, such as those mentioned by the referee, and will use that as an opportunity to discuss the broader issue of what to do when the Bayesian models give conflicting results.

We agree with reviewer 2's point that the model description terminology could be made clearer for the target eLife audience. We tried to strike a balance between introducing the reader to the conventional technical terminology used in the Bayesian data analysis necessary for understanding the model while avoiding exhaustive statistical terminology. We erred too much on the side of the latter instead of providing clear links between the model construction and experimental data. In the revised version of the paper, we will augment any technical terms with more biological language and provide a Glossary for reader reference.